Some problems that come to mind with the fiat banking system are:

• Transactions are never really "final" (unless you use cash) since banks or credit cards can "charge back" or "reverse" them at any future time simply because they take the side of your counter-party, often without even listening to your side
• The justice system can take your money at any time. If a court decides that your money may have been ill-gotten or that you owe someone else money (due to a lawsuit or divorce for example) they can unilaterally take as much as they see fit.
• The legislative system can take your money at any time. If a legislative body determines that it needs your money for some urgent need (such as to pay back national debt, as happened in Cyprus) they can unilaterally take as much as they see fit.
• The bank themselves can take your money at any time (for example if they see your business as "risky" or suspect you of making money illegitimately).
• The bank can (and virtually always does) limit your ability to access your own money. They can make up rules about how much of your money you can withdraw in a certain time and question your purposes before allowing you to withdraw large amounts.
• The banks, government agencies and potentially hackers have access to your transactions... who paid you, and who you paid and when.
• Legislative and executive bodies in the government can change the value of your money at any time by manipulating interest rates and the money supply
• Banks and governments can (and virtually always do) limit your ability to move your money across national borders. And even when they allow it, they charge banking and exchange fees, delay your transaction significantly, and subject it to increased scrutiny and record-keeping
• Banks and governments can (and do) limit your ability to transact with anyone you want. If there is someone they want to starve of funding they will prevent you from paying them.
• Banks allow a "pull" model of transaction. In other words, not only you but anyone else can initiate a payment out of your account. As long as the bank believes the transaction to be legitimate they will allow someone else to reach into your account and take your money.